Propiedades asociadas a la métrica η

Transcripción

1 Apéndice A Propiedades asociadas a la métrica η Se demuestra la relación que existe entre la métrica η con el volumen v y las longitudes l i j de las aristas de un tetraedro. La independencia de la métrica con respecto del nodo de referencia utilizado para calcularla ha quedado demostrada en la sección 2.1.1, a partir de mostrar que la matriz Jacobiana A es nodalmente invariante. En primera instancia se definen los elementos necesarios para llevar a cabo la demostración. Sea el tetraedro regular R definido por los vértices (r 0,r 1,r 2,r 3 ) de coordenadas r 0 = ( 3a/2,0,0) r 1 = (0, a/2,0) r 2 = (0,a/2,0) r 3 = ( 3a/6,0, 6a/3) que se presenta en la figura A.1. Sea T un tetraedro cualquiera definido por los vértices (t 0,t 1,t 2,t 3 ). Se indica con R a la matriz Jacobiana del tetraedro R referenciada al vértice r 0 y con T a la matriz Jacobiana del tetraedro T referenciada a su vértice t 0. Luego, R = [r 1 r 0 r 2 r 0 r 3 r 0 ] (A.1) es decir que 3/2 3/2 3/3 R = a 1/2 1/ /3, R 1 = 1 a 1/ 3 1 1/ 6 1/ 3 1 1/ / 6 Bajo la condición de que ambos T y R deben tener el mismo volumen v, es decir (A.2) v = 1 6 det(r) (A.3) 127

2 se obtiene el valor de a a = (6 2v) 1/3 (A.4) Figura A.1: Tetraedro regular R Además se denota con el escalar d i j = (t j t i ) T (t j t i ) al cuadrado de la longitud de la arista definida por los nodos i j. Relación con el volumen y longitudes de las aristas Para ello se debe desarrollar tanto la expresión del det(a(r,t)) como de la traza Tr(A(R,T)). Por definición (ver sección 2.1.1) se sabe que A(R,T) = M T (R,T)M(R,T) (A.5) = (R 1 ) T T T T R 1 (A.6) y por propiedad del determinante deta(r,t) = det(m T M) (A.7) = det(m T )det(m) (A.8) = det(m) 2 (A.9) pero dado que se impone la condición de que R y T deben tener el mismo volumen, entonces det(m) = det(t )det(r 1 ) (A.10) = 1 (A.11) 128

3 Con lo cual se concluye que det(a(r,t)) = 1 (A.12) Por otra parte T T T = d 01 (d 01 + d 02 d 12 )/2 (d 01 + d 03 d 13 )/2 (d 01 + d 02 d 12 )/2 d 02 (d 02 + d 03 d 23 )/2 (A.13) (d 01 + d 03 d 13 )/2 (d 02 + d 03 d 23 )/2 d 03 Luego de Ecs.(A.2) y (A.13) se obtiene A(R,T) = de donde (2d d 02 d 12 )/ d (d 03 + d 13 + d 23 ) 6 1(d 01 + d 02 + d 12 ) (A.14) Tr(A(R,T)) = 1 2a 2 (d 01 + d 02 + d 03 + d 12 + d 13 + d 23 ) = 3 li 2 j i, j=0,i j (A.15) (A.16) Reemplazando los valores para el det(a) y la Tr(A) en la Ec.(2.13) se obtiene η(t) = 3 3 λ 1 λ 2 λ 3 (λ 1 + λ 2 + λ 3 ) (A.17) η(t) = = a 2 3 i, j=0,i j l2 i j 6a 2 3 i, j=0,i j l2 i j (A.18) (A.19) Reemplazando el valor de a dado por la Ec.(A.4) en esta última ecuación se llega a la Ec.(2.14) 129

4 η(t) = 6(6 2v) 2/3 3 i, j=0,i j l2 i j = 12(3v)2/3 6 i=1 l2 i j (A.20) (A.21) 130

5 Apéndice B Sistema de ecuaciones diferenciales ordinarias de Taylor-Sedov En este apéndice se deriva el sistema de ecuaciones diferenciales ordinarias a partir del cual se obtienen los perfiles auto-similares que describen el comportamiento de las variables del flujo en el problema de la onda explosiva esférica o problema de Taylor-Sedov. A tal fin, se parte de las ecuaciones (7.10), (7.11) y (7.12) y se hace uso de las definiciones para las variables del flujo dadas por las ecuaciones (7.13) y sus derivadas (7.14). Para operar sobre la ecuación de continuidad (7.10) es necesario calcular las siguientes derivadas parciales ρ t ρ r u r = ρ ( ) 0 t Ω(ξ) + ρ 0 ξṙ R Ω (ξ) = ρ 0 r Ω(ξ) + ρ 1 0 R Ω (ξ) = Ṙ R U (ξ) (B.1) (B.2) (B.3) Reemplazando las ecuaciones (B.2), (B.1) y (B.3) en la Ec.(7.10) se obtiene la siguiente ( ρ 0 Ω(ξ) ρ 0 ξṙ R Ω (ξ) + ṘU(ξ) ρ Ω ) (ξ) 0Ω(ξ) + ρ (B.4) R... + Ṙ R U (ξ)ρ 0 Ω(ξ) + 2 ρ 0Ω(ξ)ṘU(ξ) = 0 ξr Multiplicando por el factor R/(ρ 0 Ṙ) y agrupando términos se llega a la Ec.(7.15). De manera similar, introduciendo las derivadas parciales 131

6 u t p r = RU(ξ) ξṙ2 R U (ξ) = ρ 0 r Ṙ2 P(ξ) + ρ 0 Ṙ 2 R P (ξ) (B.5) (B.6) en la ecuación de momento (7.11) se llega a que RU(ξ) ξṙ2 R U (ξ) + Ṙ2 R U(ξ)U (ξ) + 1 ρ 0 Ω(ξ) [ρ 0Ṙ2 Ṙ 2 ] P(ξ) + ρ 0 R P (ξ) = 0 (B.7) Multiplicando por el factor RΩ(ξ)/(Ṙ 2 ) se obtiene la ecuación (7.16). Finalmente, si se tiene en cuenta que el cuadrado de la velocidad del sonido c s se expresa en función de las definiciones para p y ρ dadas por las ecuaciones (7.13) como c 2 s = γṙ 2 P(ξ)/Ω(ξ) (B.8) y que la derivada parcial de la presión respecto del tiempo está dada por p t = ρ Ṙ 0Ṙ2 P(ξ) + 2ρ 0 Ṙ RP(ξ) 3 ξρ 0 R U (ξ) (B.9) Reemplazando las ecuaciones (B.8), (B.9), (B.6), (B.2) y (B.1) en la ecuación de la energía (7.12) se obtiene la siguiente ρ 0 Ṙ 2 Ṙ P(ξ) + 2ρ 0 Ṙ RP(ξ) 3 ξρ 0 R P (ξ) + ṘU(ξ) [ρ 0Ṙ2 Ṙ 2 ] P(ξ) + ρ 0 R P (ξ) γṙ 2 P(ξ) [ ] Ṙ ρ 0 Ω(ξ) ξρ 0 Ω(ξ) R Ω (ξ) γṙṙ 2 P(ξ)U(ξ) [ ρ Ω ] (ξ) Ω(ξ) 0Ω(ξ) + ρ 0 = 0 (B.10) R Multiplicando por el factor R/(ρ 0 Ṙ 3 ) y agrupando términos se llega a la ecuación (7.17). 132

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